Reflection implies the SCH

Saharon Shelah

doi:10.4064/fm198-2-1

Reflection implies the SCH

Saharon Shelah^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

17 Scopus citations

Abstract

We prove that, e.g., if μ > cf(μ) = N₀ and μ > 2^N0 and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality N₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality N₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

Original language	English
Pages (from-to)	95-111
Number of pages	17
Journal	Fundamenta Mathematicae
Volume	198
Issue number	2
DOIs	https://doi.org/10.4064/fm198-2-1
State	Published - 2008

Keywords

Pcf
Reflection
Set theory
Singular cardinal hypothesis
Stationary sets

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10.4064/fm198-2-1

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N2 - We prove that, e.g., if μ > cf(μ) = N0 and μ > 2N0 and every stationary family of countable subsets of μ+ reflects in some subset of μ+ of cardinality N1, then the SCH for μ+ holds (moreover, for μ+, any scale for μ+ has a bad stationary set of cofinality N1). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

AB - We prove that, e.g., if μ > cf(μ) = N0 and μ > 2N0 and every stationary family of countable subsets of μ+ reflects in some subset of μ+ of cardinality N1, then the SCH for μ+ holds (moreover, for μ+, any scale for μ+ has a bad stationary set of cofinality N1). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

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KW - Reflection

KW - Set theory

KW - Singular cardinal hypothesis

KW - Stationary sets

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JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

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Reflection implies the SCH

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Keywords

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